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Abstract: Given an equivalence class in the measure algebra of the Cantor space, let be the set of points having density in . Sets of the form are called -regular. We establish several results about -regular sets. Among these, we show that -regular sets can have any complexity within (), that is for any subset of the Cantor space there is a -regular set that has the same topological complexity of . Nevertheless, the generic -regular set is -complete, meaning that the classes such that is -complete form a comeager subset of the measure algebra. We prove that this set is also dense in the sense of forcing, as -regular sets with empty interior turn out to be -complete. Finally we show that the generic does not contain a set, i.e., a set which is in . [Copyright &y& Elsevier]